What is the importance of finding away to descirbe prime numbers in relation to both themselves as well as other numbers?

I am afraid the importance is purely intellectual - at least so far. One never knows when a mathematical pursuit will have some practical value. But the reason people do these things is because it is interesting (at least to those who do them!) and is important to other mathematicians. The same thing applies to physicists, too, although there may be more frequent spin-off benefits from some discovery. It may not be readily apparent that there is any practical value in finding the top quark or Higgs boson or in figuring out whether black holes radiate, but that is not why these things are pursued.

The relation of primes to composites is all-important. Without knowledge of the unique prime factorization of a number, we would be nowhere in Number Theory. As a result, we would not have public key encryption, and Ebay would not exist.

Of course, it was thought - not too long ago - that Number Theory would have no practical application.

To expand further, everytime you access a web site whose URL begins with "https:" (such as your on-line banking, credit card transaction site, etc.), you are using security protocols made possible by Public Key Encryption (PKE) and the unique properties of Prime Numbers. A quick description of PKE mathematics and its use of primes can be found here:http://world.std.com/~franl/crypto/rsa-guts.html

The relation of primes to composites is all-important. Without knowledge of the unique prime factorization of a number, we would be nowhere in Number Theory. As a result, we would not have public key encryption, and Ebay would not exist.

Of course, it was thought - not too long ago - that Number Theory would have no practical application.

Although I wasn't aware that prime number theory forms an essential part of encryption theory, that would provide good support for an argument that pure science and mathematics can have a practical spin-off, and so should be supported economically. But I would hate to think that the absence of practical value should deter anyone from the intellectual pursuit of knowledge, or from supporting it economically. Peer review is the proper and best way to ensure that a particular pursuit is worthwhile, not practical usefulness.

The RSA algorithm, based on prime numbers, is beautiful. This is at it's heart:

[tex]p^e\equiv c Mod (n)[/tex]
[tex]c^d\equiv p Mod (n)[/tex]

To learn and understand these two equations, should cause anyone to acquire an appreciation of prime numbers. Imagine looking at a 512-digit number (a real integer, not just some digits strung together) and thinking, "there's a real sentence in there" and without the decryption exponent, no one on earth can figure out what it is!